Static-arbitrage lower bounds on the prices of basket options via linear programming

Javier Peña, Juan C. Vera, Luis F. Zuluaga
2010 Quantitative finance (Print)  
We show that the problem of computing sharp upper and lower static-arbitrage bounds on the price of a European basket option, given the prices of other similar options, can be cast as a linear program (LP). The LP formulations readily yield super-replicating (subreplicating) strategies for the upper (lower) bound problem. The dual counterparts of the LP formulations in turn yield underlying asset price distributions that replicate the given option prices, and the bound on the new basket
more » ... price. In the special case when the given option prices are those of vanilla options on the underlying assets, we show that the LP formulations admit further simplifications. In particular, for the upper bound problem we derive closed-form formulas for the basket's price bound, and for the corresponding superreplicating strategy. In addition, our LP approach admits efficient modeling of additional features such as basket options with negative weights, bid/ask spreads, transaction costs, and diversification constraints. We provide numerical experiments to illustrate some of our results. We note that the problems (U) and (DU) are semi-infinite linear programs. However, as Proposition 2 below shows, (DU) can be recast as a linear program. For ease of exposition, we shall use the following notational conventions. Let W denote the r × n matrix whose j-th row is the vector w j for j = 1, . . . r and let W the (r + 1) × n matrix whose j-th row is the vector w j for j = 0, 1, . . . r. Also, let K denote the vector K 1 , . . . , K r T and K = K 0 , K 1 , . . . , K r T . Given v ∈ R I for some finite index set I, and J ⊆ I, let v J ∈ R J denote the vector formed by the entries v j with j ∈ J. Likewise, if the rows of M are indexed by I and J ⊆ I, let M J denote the matrix formed by the rows of M indexed by J. Also, we shall write J c as a shorthand for I \ J. The larger set I will typically be of the form {0, 1, . . . , r} or {1, . . . , r}, for some positive integer r. Given J ⊆ {0, 1, . . . , r}, define and let J = {J ⊆ {0, . . . , r} : P J = ∅}. m i=0 J i = {1, . . . , n} , J i ∩ J j = ∅ for i = j . Given J ∈ P(n, m), define Hence from the last inequality in (35) and the fact that γ m,J = γ m J m we get m i=0   i j=0 y j J Adding all of these inequalities and rearranging terms, we get Since −c ≤ −e T ξ, we get (34). Now assume that (34) holds. For = 1, . . . , n let This choice of ξ ensures that the first four constraints in (37) hold. Let J ∈ P(n, m) be such that Furthermore, from (34) (applied to the partition J) we have Hence the last constraint in (37) holds as well. This completes the equivalence between (33) and (34). 2 Proof of Theorem 11. The equivalence between (18) and (19) follows from Theorem 18. The latter in turn yields the equivalence between (12) and (20) . Finally, by linear programming duality and Proposition 1 it follows that (21) yields the optimal value of (11). 2
doi:10.1080/14697680902956703 fatcat:l23twf5snnea3dc2xqs5fuyokq